2.001 - MECHANICS AND MATERIALS I
Lecture # 13
10/25/2006
Prof. Carol Livermore
Recall from last time:
σxx
[σ ] = σyx
σzx
σxy
σyy
σzy
Diagonal terms are normal stresses.
Off diagonal terms are shear stresses.
Not as bad as it seems:
1. Linearity
2. Superposition
EXAMPLE: Uniaxial Stress
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σxz
σyz
σzz
.
σ
0
[σ ] = 0
0
0 0 0 0 .
0 0
EXAMPLE: Hydrostatic Stress
−p 0
0
[σ ] = 0 −p 0
0
0 −p
.
Note: p is negative (compressive stresses are negative by convention)
EXAMPLE: Plane Stress
σa
[σ ] = τ
0
2
τ
σb
0
0
0
0
.
EXAMPLE: Biaxial Plane Stress
σa
[σ ] = 0
0
0
σb
0
0
0
0
.
EXAMPLE: Shear Plane Stress
0
[σ ] = τ
0
τ
0
0
0
0
0
.
Note: The stress at a material point that you see, and its magnitude, depends
on the orientation of the coordinates relative to your loading.
In other words: the state of stress is a function of the chosen coordinate system.
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This is called a ”stress transformation”
Equilibrium:
−σxx dydz + (σx x +
Fx = 0
∂σyx
∂σxx
dxdydz = 0
dx)dydz − σyx dxdz + (σyx
∂x
∂y
So:
∂σxy
∂σxx
+
∂x
∂y
Fy = 0
∂σyy
∂σxy
+
∂y
∂x
M
(σxy +
∂σxy
dx
dx
dy
∂σyx
dy
dx) dydz +σxy dydz −σyx dxdz −(σyx +
dy)dxdz
=0
∂x
2
2
2
∂y
2
σxy − σyx +
∂σxy dx ∂σyx dy
−
=0
∂x 2
∂y 2
For dx, dy → 0:
σxy = σyx
This shows that the stress tensor (σ) is symmetric.
This was derived in plane stress (2-D) but it can be extended to 3-D. Here are
the results:
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σxy = σyx
σyz = σzy
σxz = σzx
So:
σxx σxy
[σ ] = σyx σyy
σzx σzy
σxz
σyz
σzz
.
Note: 6 independent stress terms due to equilibrium. The stress tensor is symmetric.
Strain:
Uniaxial:
du(x)
dx
δ
=
L
(x) =
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u(x, y, z) = ux (x, y, z)î + uy (x, y, z)ĵ + uz (x, y, z)k̂
Case 1:
u(x, y, z) = aî
No strain. Rigid body translation.
Case 2:
u9x, y, z) = aî + bĵ
No strain. Rigid body translation.
Case 3:
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Isotropic Deformation: Stentches evenly so this is linear.
u(x, y, z) =
δx ˆ
i
L
dux
δ
=
dx
L
Define:
xx =
dux
, Normal Strain (x face in x direction)
dx
Case 4:
u(x, y, z) =
a+
δ
x î
L
dux
δ
=
dx
L
Note: Rigid body translation does not effect the strain.
xx =
Case 5:
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δ
y ĵ
L
duy
δ
=
=
dy
L
u(x, y, z) =
yy
Case 6:
δ1
δ2
xî + y ĵ
L
L
dux
δ1
=
=
dx
L
duy
δ2
=
=
dy
L
u(x, y, z) =
xx
yy
Case 7: Shear Strain
u(x, y, z) =
δ
y ĵ
L
∂ux
δ
= = tan θ ≈ θ
∂y
L
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Define:
γxy = θ
u(x, y, z) =
δ
xĵ
L
δ
∂uy (x)
= =θ
∂x
L
γyx = θ
γyx =
dux
duy
+
dy
dx
Define:
xy = yx
1 ∂ux
∂uy
=
+
2 ∂y
∂x
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